Boundary Labeling for Rectangular Diagrams

Given a set of $n$ points (sites) inside a rectangle $R$ and $n$ points (label locations or ports) on its boundary, a boundary labeling problem seeks ways of connecting every site to a distinct port while achieving different labeling aesthetics. We examine the scenario when the connecting lines (leaders) are drawn as axis-aligned polylines with few bends, every leader lies strictly inside $R$, no two leaders cross, and the sum of the lengths of all the leaders is minimized. In a $k$-sided boundary labeling problem, where $1\le k\le 4$, the label locations are located on the $k$ consecutive sides of $R$. In this paper, we develop an $O(n^3\log n)$-time algorithm for 2-sided boundary labeling, where the leaders are restricted to have one bend. This improves the previously best known $O(n^8\log n)$-time algorithm of Kindermann et al. (Algorithmica, 76(1):225-258, 2016). We show the problem is polynomial-time solvable in more general settings such as when the ports are located on more than two sides of $R$, in the presence of obstacles, and even when the objective is to minimize the total number of bends. Our results improve the previous algorithms on boundary labeling with obstacles, as well as provide the first polynomial-time algorithms for minimizing the total leader length and number of bends for 3- and 4-sided boundary labeling. These results settle a number of open questions on the boundary labeling problems (Wolff, Handbook of Graph Drawing, Chapter 23, Table 23.1, 2014).